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1. (i) Predoeite geouretrijski i navedite znadeu.je tj.f k:.g)rlrd,g, gclje je
/ neka pozitivrra funkcija i D podru<I .ie t r'[ ravnini. (1 bocl)
{z-4u,&)9f *r*'71dxd"g i**q
- , |
-L
*1
I
4/L q"a
(iii) Izradunajte teZi5te(1 bod)
a n t I r
Y, \ Jr4{2, r ) dri ,a I* * =
(zx"+
=3,3 \
*7^d ̂ Y " hq''o't'
2 r >4^*J 7'a u't i 4"x
(zx r,93 \a* :
\S &*.y4!ttr= f [l-U*.r')^il*Dz-a\ V,*, \= if exb, €)\ l'"L- &z
: fi,*,*i-',,f l Jd'L
t - 5=-3 ? *3 3
tt'!V'\
Brqt
: "'AcLt)t u J4t7+"
_ 3ql. 476
ry z't36t
t 46oA J "q+" fu*huio
(ii) Izrai:unaite irrtegral iz (i) ako je f (*,U):2t;*,U, i D za<lano s2 1 n < 3, -3 S l) < -2. Slikal Objasnite znai:enje tog integralanko je / funkcija gustoie rlase. opisite rijedirna rnzdiobu rrase.(1 bod)
G t, -t * o c'\ Auail4 fowe<v1
5e ootLosu',1 W byo L
o o) ,^ * g ̂ v- ot<Lq I "U9),
',
" d,";"1.v qwqq ̂ tuk.u - u hlu^'*ar,t{)ry)oF ,.}-,.
t-r,*\:i, "*"*Y"^u'r.rrrJ"J-\;;:} vvh'ayn
-2
t ̂ - i z n""-L :dt
l/t'lu e
V'a\ tJ \ D
3
Iu
( T'r tX3
1(\
3 L l t
7'
?to €*)
5* EJ
t *tQI W
(rr,Ar,) za fuukci.ju gusto(;e nrase iz (ii).
Wr"t*)Jv;17'r/ z
ffi=# j lietrx)')d:l ,t*5 =-z
z ? ar'6ffi t " E') l_ d*Tq
)V\'W t:-j
Hr= { v'Yl. Ja"tz
331
'{+ I68
12 t
)l'Z
3 z" ry,v
3. (i) Napisite op6u linearnu diferencijalnu jed[adzbu prvog reda.
Objasnite podjehr na hornogene i nehornogeue, (1 bod) 6
bL hc*l*=yk) kuo 6 &=o ('qt4(\'ukt1u)n ^ i
!r$ f,it,"utal( 0 \'e # &Adt"$"a' 'l"w"grc u<i
y,= S h '* n" e")'A =- o )\ r d:c
t iarr"€ A' A^?RA"W#L,L4
(ii) Izdvojite linearne jednadZbe od nelinearnih, a rne<lu lineantima
izdvojite hornogene od rrehomogenih' Objasnite! (1 bod)
a) Srsirr !) * Y' :2, lr) 2r/A a Y' ,rr* ;9) b
'
n.....-<:) ye" *y'lnr:2 , rl) zJi +y {/cos;r.
q) U<-h.{a-e a\. r,.r"c, 76 ,j a>', q 3
b) MeJ4'v€arr,nrs z\U"q z , L,r>c): *#,
c) kunutvrt{ '" Z'* ff*"J= 6* ,f,*r= h*rm?Lrr. 'n"Jaa4
d\ho*e^YLLa,oo 3t- *- ' I =
# )
g c,: * d:,-r. tr idr'
. r tl^G()= ir* )
M€hu'u""rju t4Q '
(iii) Kako Hj::.Jn'iffi';,ffiljHrfi ,l':;:":',;"i?,;jjioun.'
.I, V-hoA. r,1+,
' t d"'r;';"' X::^*^- Jbl" .So-
{'tt)'J=u'
Elvie *d( & A= c..utl ., Z etF s uT' 'onU
6rwvtrr{rpa^* 1fu ",0tr1t *pl 'l^ kl ,
J ; &.,L""L "l*ruh, *t "Y n if 't{-€ fu'^'^YtY W i'' u
A = c;'"eu^)', ccz) /^ *l,ut\ (yU
v wyzu;* ho 4),,r/kq\e ?qr^.r'Jt 'ni [u /-'"k'"ttd
?n*ty ,Ati1q-,e-"a-o /"w*ir u WU^di Y" ;;* * b '=
$'
{,* N ,r"^*, ,t #i;{::.: ",:*'=: t'a-' t+r*>=-Q^r-i
p[d!,,t.a \1"ouu;,t"al g- fu'-r ? ;: o fe $ = C e *"tr
"CeR
?' b =- # = t*€t ,,."5_t
"' t 2q >e )o i t- x/ c
L u ;; t/.- b-= zi,,r,r*= +, f,)ttr,t'{".
z. v,u.p!., I = #,' *;,:,' -#, * HH,! 8r:," v:F
",.fu ' t[..t,)z.
4. (i) Zapi5ite i pojasnite Cauchyev problem prvog i drugog reda.(1 bod)
, r , . - . ^ r , f " t . , , , . . / , " , : iw lwd'kC . p r-1,,1tlnn
"JL' w zL^ i 9'a-'9v d'tk ty''(My!-J
{ , wd}, t T" te*'laetr' ;; f ,'
,: *,ff ='o,:;; ":ff{, ;,w*; i r I ̂ ; 3,ffi n
r^^v^ l r , , ' uvxr l l /Av*au t , 0 n , - ) . 7 ,V. b c*i-- b, ( r'rdu, uwd
\/,!,7lir,o,r,,l-_. (&{ +"r("'z'vt"&)\ Til;
n='i-" ) P'ze*nn; vutr'th
4l:{,T.?: ,.Y9:11 \*."fr'ar= \'/o d
' (ii) Rije5ite Cauchyev problem prvog reda
tz z ( r i ii' .*"n,'')J:n#;'ilyl ^t tt "t t*oti /'z
* eZ +u. ) FeV tr kP3=x-
r, - 9t*g &dt'{)= z
" = <
\ 2 -
?WV C*Lat\t ('J&ua
oa.E.Uil (: c" u" gt*,b=?=A
--z--
(iii) U Cauchyevom problemu titranja
y" + w2y - o, a(o) : A, y'(o) : o
objasnite znadenje 'A,'a' , 'A" ,w2, A le podetnih uvjeta. Napi5ite rje-
Senje i komentirajte. (1 bod)
{ - rf}''1v"r,r g
b - V'b, U ie^a*; q
h'= H btuu* int';Q
h,r= 4]! aku.hrx-uyd-ff2
- Wwuc,lu tr'{"y*^ daL;+^*ap- ie"*t; ("M1I e2,\' Le
lr to) y^" fe/^+ brq'*
t, v>*iQ
tr\.oitr.ty o" A-= A' ctSL':A )
' r -+ a;T
p€ v o-o, .i'trqs t
= xar("t)Iz1f
A
h to)
\
I
>3C.U
g
5. (i) Zapi5ite precizno linearnu diferencijalnu jednadZbu 2. reda s kons-tantnim koeficijentima. (1 bod)
{ "* PZt * Z 7 = Eio) ) ? 11-€lF Oe*l^, LrclYrzr )t;) $o*uh c,)i:*
t t
NV* 41 &= o t Pn'*rt ;r"e ( hol*tolfz+t< i
3 "t 9b' + 2'Z= o ) i 'qd 4" '1,( € ,& o&gE uq
(ii) OpiSite kako se dobije opie rje5enje homogene diferencijalne jed-
nadZbe iz ( i ) . (1 bod)
1y&* p rr 2-=o Vor'*,A"txv?'14" i*A FA^Ut!$r'nt^n/z\e ott l , j tAr", tz ( i))
f,r 1T- tfYrztlu Wv*'ldturt &q"e yaueh'U<
(D)
(r I ,[tc .oIorL(
A"k*I "t"t"d't
NVe
n\ 4 rZ -
E/4 arl? w$*t i q+vz s'-J* #
5-r.{ *J't W = c4 g'a*+ C.e''?* C't,C"elK
F "n; Q = r ( d u'o fti"k' u*^)" u'v'z'1't )
11 g'F( e vpcY'cg{ t = cn ewo cz x e'*o'' C 1'C'€lt
g.t ( n e;cM'|, d'^fM> -'t-\9 * 9-t4J-' t' Yn
J,*pt ) 11 +o ) c't''st^ J4 op/e nlt("u^l(
! = ed* ('n &*Qt*\* cL h"'^(P"1) ' 'n'c'elF
(D\
(iii) Rije5ite diferencijalne jednadZbe:
A" + 6y' + 9A : 0, A" - 4A' * 73y : 0, A" * 2A' - 89 : 0. (1 bod)
*t\v (o1tt9 4= o
f '*G*ts -Q
( yasv,6l$- M"')f t= fz= *3
e yle lqc'tir :3 {
j l i
\= C, Q') CLxe )
cttCzelF
k"- tr)rtnl io
r t - 4r"YlJzo( lc+"t'r"/'$ Yfu)firz= \: {A
tlJe$orL+..r-J =t Qno"lti- ifu)
f . t= - t t { -7=L
o p /e qeiexi c "'
f , : cad\** cz "t* )€
3 2 - l 3u'
*=7 " ) P=3
A-- e'*(ct Qs (.')t
C4rc, el?
C. At(: t)c'{\cz-47